You can use the following syntax to check your answers. Note the answers you get in R will not be EXACTLY the same you get by hand but they should be pretty close.

If your \(x = 542\) (ie the number of “successes” and your \(n = 3611\), this is how you can have R calculate a 90% Confidence Interval for you.

binom.test(x = 768, n = 1024,  conf.level = .99)[["conf.int"]]
## [1] 0.7135318 0.7841063
## attr(,"conf.level")
## [1] 0.99

25.

  1. p hat = x/n = 542/3611 = 0.150
  2. np(1-p) = 3611 x 0.150 x 0.850 = 460.40 >= 10, and the smple is less than 5% of the population.
  3. 0.150 +- 0.0098, lower bound: 0.1402 uppoer bound: 0.1598
  4. We are 90% confident that the proportion of adult Americans 18 years and older who have used their smartphones to make a purchases is between 0.1402 and 0.1598.

26.

  1. p hat = 0.430
  2. np(1-p) = 1153 x 0.430 x 0.570 = 282.60 >= 10, and the sample is less than 5% of the population.
  3. 0.430 +- 0.0286, Lower bound: 0.4014 Upper bound: 0.4586
  4. We are 95% confident that the proportion of workers and retirees in the U.S. 25 years and older had less than $10,000 in savings is between 0.4014 and 0.4586.

27.

  1. p hat = 0.519
  2. np(1-p) = 1003 x 0.519 x 0.481 = 250.39 >= 10, and the sample is less than 5% of the population.
  3. 0.519 +- 0.0309, Lower bound: 0.4881 Upper bound: 0.5499
  4. Yes; it is possible that the population proportion is more than 60%, because it is possible that the true proportion is not captured in the confidence interval. It is not likely.
  5. 0.481 +- 0.0309, Lower bound: 0.4501 Upper bound: 0.5119

28.

  1. p hat = 0.75
  2. np(1-p) = 1024 x 0.75 x 0.25 = 192 >= 10, and the sample is less than 5% of the population.
  3. 0.75 +- 0.0348, Lower bound: 0.7152 Upper bound: 0.7848
  4. Yes it is possible that the population proportion is less than 70%, becauseit is possible that the true proportion is not catured in the confidence interval. It is not likely.
  5. 0.25 +- 0.0348, Lower bound 0.2152 Upper bound: 0.2848

29.

  1. p hat = 0.540
  2. np(1-p) = 1748 x 0.540 x 0.460 = 434.20 >=10, and the sample is less than 5% of the population.
  3. 0.540 +- 0.0196, Lower bound: 0.5204 Upper bound: 0.5596
  4. 0.540 +- 0.0307, Lower bound: 0.5093 Upper bound: 0.5707
  5. Increasing the level of confidence widens the interval.